Consider an insulated rigid container of gas separated into two
halves by a heat conducting partition so the temperature of the gas
in each part is the same. One side contains air, the other side
another gas, say argon, both regarded as ideal gases. The mass of
gas in each side is such that the pressure is also the same.

The entropy of this system is the sum of the entropies of the two
parts:
.
Suppose the partition is taken away so the gases are free to diffuse
throughout the volume. For an ideal gas, the energy is not a
function of volume, and, for each gas, there is no change in
temperature. (The energy of the overall system is unchanged, the two
gases were at the same temperature initially, so the final
temperature is the same as the initial temperature.) The entropy
change of each gas is thus the same as that for a reversible
isothermal expansion from the initial specific volume
to the
final specific volume,
. For a mass
of ideal gas, the
entropy change is
. The entropy change
of the system is

(7..1)

Equation (7.1) states that there is an entropy
increase due to the increased volume that each gas is able to
access.

Examining the mixing process on a molecular level gives additional
insight. Suppose we were able to see the gas molecules in different
colors, say the air molecules as white and the argon molecules as
red. After we took the partition away, we would see white molecules
start to move into the red region and, similarly, red molecules
start to come into the white volume. As we watched, as the gases
mixed, there would be more and more of the different color molecules
in the regions that were initially all white and all red. If we
moved further away so we could no longer pick out individual
molecules, we would see the growth of pink regions spreading into
the initially red and white areas. In the final state, we would
expect a uniform pink gas to exist throughout the volume. There
might be occasional small regions which were slightly more red or
slightly more white, but these fluctuations would only last for a
time on the order of several molecular collisions.

In terms of the overall spatial distribution of the molecules, we
would say this final state was more random, more mixed, than the
initial state in which the red and white molecules were confined to
specific regions. Another way to say this is in terms of
``disorder;'' there is more disorder in the final state than in the
initial state. One view of entropy is thus that increases in entropy
are connected with increases in randomness or disorder. This link
can be made rigorous and is extremely useful in describing systems
on a microscopic basis. While we do not have scope to examine this
topic in depth, the purpose of this chapter is to make plausible the
link between disorder and entropy through a statistical definition
of entropy.